The cosmological constant problem Steven Weinberg Theory Group, Department of Physics, University of Texas, Austin, Texas 7871Z Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller than estimated in modern theories of elementary particles. After a brief review of the history of this prob- lem, five different approaches to its solution are described. CONTENTS II. EARLY HISTORY After completing his formulation of general relativity I. Introduction 1 in 1915—1916, Einstein (1917)attempted to his new II. Early History 1 apply III. The Problem 2 theory to the whole universe. His guiding principle was IV. Supersymmetry, Supergravity, Superstrings 3 that the universe is static: "The most important fact that V. Anthropic Considerations 6 we draw from experience is that the relative velocities of A. Mass density 8 the stars are very small as compared with the velocity of 8. Ages 8 light. " No such static solution of his original equations C. Number counts 8 could be found (any more than for Newtonian gravita- VI. Adjustment Mechanisms 9 VII. Changing Gravity 11 tion), so he modified them by adding a new term involv- VIII. Quantum Cosmology 14 ing a free parameter A., the cosmological constant: IX. Outlook 20 — — Acknowledgments 21 R„——,'g R A,g„= 8e GT„ (2.1) References 21 Now, for A, &0, there was a static solution for a universe filled with dust of zero pressure and mass density As I was going up the stair, wasn't I neet a man who theve. (2.2) He wasn't there again today, 8+6 he'd I wish, I wish stay away. Its geometry was that of a sphere S3, with proper cir- Hughes Mearns cumference 2m.v, where r = 1/VSmpG (2.3) I. INTRODUCTION so the mass of the universe was Physics thrives on crisis. We all recall the great pro- gress made while finding a way out of various crises of M=2mr p= —k ' 6 (2.4) the past: the failure to detect a motion of the Earth 4 Hubble' through the ether, the discovery of the continuous spec- In some popular history accounts, it was s Ein- trum of beta decay, the ~-0 problem, the ultraviolet discovery of the expansion of the universe that led divergences in electromagnetic and then weak interac- stein to retract his proposal of a cosmological constant. tions, and so on. Unfortunately, we have run short of The real story is more complicated, and more interesting. crises lately. The "standard model" of electroweak and One disappointment came almost immediately. Ein- be- strong interactions currently faces neither internal incon- stein had been pleased at the connection in his model sistencies nor conflicts with experiment. It has plenty of tween the mass density of the universe and its geometry, Mach's loose ends; we know no reason why the quarks and lep- because, following lead, he expected that the tons should have the masses they have, but then we know mass distribution of the universe should set inertial frames. was his no reason why they should not. It therefore unpleasant when friend de Perhaps it is for want of other crises to worry about Sitter, with whom Einstein remained in touch during the that interest is increasingly centered on one veritable war, in 1917 proposed another apparently static cosmo- crisis: theoretical expectations for the cosmological con- logical model with no matter at all. (See de Sitter, 1917.) Its line element the same coordinate as de stant exceed observational limits bP some 120 orders of (using system magnitude. ' In these lectures I will first review the histo- Sitter, but in a difterent notation) was the various 1 ry of this problem and then survey attempts dv = [dt that have been made at a solution. cosh Hv dr— H tanh Hr(dO —+ sin Odg )], *Morris Loeb Lectures in Physics, Harvard University, May (2.5) 2, 3, 5, and 10, 1988. For a good nonmathematical description of the cosmological 2The notation used here for metrics, curvatures, etc., is the constant problem, see Abbott (1988). same as in W'einberg (1972). Reviews of Modern Physics, Vol. 61, No. 1, January 1989 Copyright 1988 The American Physical Society Steven Weinberg: The cosmological constant problem with H related to the cosmological constant by III. THE PROBLEM H =&A, /3 (2.6) Unfortunately, it was not so easy simply to drop the and p=p =0. Clearly matter was not needed to produce cosmological constant, because anything that contributes inertia. to the energy density of the vacuum acts just like a At about this time, the redshift of distant objects was cosmological constant. Lorentz invariance tells us that being, discovered by Slipher. Over the period from 1910 in the vacuum the energy-mornenturn tensor must take to the mid-1920s, Slipher (1924) observed that galaxies the form (or, as then known, spiral nebulae) have redshifts — & T„.&= (p&g„. (3.1) z = b, A, /A, ranging up to 6%, and only a few have blue- shifts. Weyl pointed out in 1923 that de Sitter's model (A minus sign appears here because we use a metric would exhibit such a redshift, increasing with distance, which for flat space-time has goo= —1.) Inspection of because although the metric in de Sitter's coordinate sys- Eq. (2.1) shows that this has the same efFect as adding a tem is time independent, test bodies are not at rest; there term 8m G (p ) to the effective cosmological constant is a nonvanishing component of the afBne connection X„=X+8~G(p) . (3.2) I « = —H sinhHr tanhHr (2.7) Equivalently we can say that the Einstein cosmological giving a redshift proportional to distance constant contributes a term A, /8mG to the total efFective vacuum energy z=Hr for Hr (&1 . (2 8) =&p&+X/8 G=A, /8 G . (3.3) In his influential textbook, Eddington (1924) interpreted p , Slipher's redshifts in terms of de Sitter's "static" A, A crude experimental upper bound on ,& or pz is pro- universe. vided by measurements of cosmological redshifts as a But of course, although the cosmological constant was function of distance, the program begun by Hubble in the needed for a static universe, it was not needed for an ex- late 1920s. The present expansion rate is today estimated panding one. Already in 1922, Friedmann (1924) had de- as scribed a class of cosmological models, with line element 1 dR (in modern notation) =Ho =50—100 km/sec Mpc R dt now 2 2 dr =dt — +r (d6 + sin Hdy ) ' ' dr R(t) —,— . 1 —kr =( 1)X10 /yr (2.9) Furthermore, we do not gross effects of the curvature of the universe, so very roughly These are comoving coordinates; the universe expands or ik i/R'„.„SH', . contracts as R (t) increases or decreases, but the galaxies keep fixed coordinates r, o, y. The motion of the cosmic Finally, the ordinary nonvacuum mass density of the scale factor is governed by an energy-conservation equa- universe is not much greater than its critical value tion — & Ip (p I -3H,'/8~G . 2 dR = —k+ —'R (8m Gp+ A, ) . (2 10) Hence (2.10) shows that dt The de Sitter model is just the special case with k =0 and or, in physicists' units, p=O; in order to put the line element (2.5) in the more general form (2.9), it is necessary to introduce new coor- ipvi 510 g/cm =10 GeV (3.4) dinates, A more precise observational bound will be discussed in ' t'= t —H ln coshHr, Sec. V, but this one will be good enough for our present purposes. r'=H ' exp( Ht) sinhHr, — (2.11) As everyone knows, the trouble with this is that the en- ergy density (p) of empty space is likely to be enormous- ly larger than 10 GeV . For one thing, summing the and then drop the primes. However, we can also easily zero-point energies of all normal modes of some field of find solutions with A, =O and 0. Pais (1982) expanding p ) mass m up to a wave number cutoff A)) m yields a vacu- quotes a 1923 letter of Einstein to Weyl, giving his reac- um energy density (with fi =c = 1 ) tion to the discovery of the expansion of the universe: "If there is quasi-static then with the JA4vrk dk 1 2 A no world, away ( ) ~k2+ (3.5) cosmological term&" (2m)' 2 16~' Rev. Mod. Phys. , Vol. 61, No. 1, January 1989 Steven Weinberg: The cosmological constant problem If we believe general relativity up to the Planck scale, of quantum Auctuations. As we have seen, the zero-point then we might take A=(SmG) ', which would give energies themselves gave far too large a value for & p ), so Zeldovich assumed that these were canceled by A, /Sn. G, =2-"~-'G-'=2X 10" GeV'. (3.6) &p) leaving only higher-order effects: in particular, the gravi- in Quc- & tational force between the particles the vacuum But we saw that ~ p) +1,/ SAG~ is less than about 10 GeV, so the two terms here must cancel to better tuations. (In Feynman diagram terms, this corresponds than 118 decimal places. Even if we only worry about to throwing away the one-loop vacuum graphs, but keep- zero-point energies in quantum chromo dynamics, we ing those with two loops. ) Taking A particles of energy would expect &p) to be of order AocD/16m, or 10 A per unit volume gives the gravitational self-energy den- GeV, requiring I,/SmG to cancel this term to about 41 sity of order decimal places.